Dignity is someone's self-respect or elevation of character. It is a measure of nobility. Considering what the word means, I think we watch George grow considerably throughout the book, but most specifically with how he deals with Lennie.
In chapter 2, George lies to the boss about Lennie so that he'll keep the two of them. When Lennie recognizes the lie he says,
"I wasn't kicked in the head with no horse was I, George?"
"Be a damn good thing if you was," George said viciously. "Save ever'body a hell of a lot of trouble."
Who says that to a friend? or even an enemy? George's character here is completely brutal and demonstrates total selfishness.
In the middle of the novel, George allows Lennie to stick up for himself when Curley tries to fight him,
Get 'im Lennie!
took minutes for George to say while watching Lennie get beat in the face. George knew Lennie's strength would hurt Curley and maybe cost them their jobs, but wasn't going to watch Lennie just get hurt. This was a parenting decision George executed brilliantly.
By the end of the novel when George knows the only honorable and noble thing he can do for himself and his friend and others he comes in contact with is to put Lennie to his Final Rest, George demonstrates compassion. When Lennie said,
"I done another bad thing."
"It don't make no difference."
This relieved Lennie of his guilt. It takes a man of true dignity to release a friend from such consequences as Lennie's actions deserved.